Quant Lesson 13: Probability • GMATCourse.mba

1

Basic Probability Framework

$P(E) = \dfrac{\text{Favorable outcomes}}{\text{Total possible outcomes}}$

Always: $0 \leq P(E) \leq 1$. Impossible: P = 0. Certain: P = 1.

Complementary Probability

$P(E) + P(\text{not } E) = 1$

$\Rightarrow P(E) = 1 - P(\text{not } E)$

Use this when "at least one" or "not all" questions arise — compute the complement instead.

2

AND, OR, and Compound Events

Independent Events (AND)

$P(A \text{ and } B) = P(A) \times P(B)$

Events that don't affect each other: flip a coin twice, roll two dice.

Mutually Exclusive Events (OR)

$P(A \text{ or } B) = P(A) + P(B)$

Events that can't both happen: rolling a 2 or a 5 on one die.

OR with Overlap (General Addition Rule)

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Use when events can occur simultaneously (e.g., drawing a heart OR an ace from a deck).

3

Conditional Probability & Dependent Events

$P(A|B) = \dfrac{P(A \text{ and } B)}{P(B)}$

P(A given B already occurred)

Dependent Events — Without Replacement

Bag: 4 red, 6 blue. Draw 2 without replacement.

P(both red) = $\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$

After first draw, denominator decreases by 1.

4

Expected Value & "At Least One"

"At Least One" Strategy

P(at least 1) = 1 − P(none)

Far easier to compute P(none) and subtract from 1.

Example

P(at least one head in 3 flips) = 1 − P(all tails)

= $1 - (\frac{1}{2})^3 = 1 - \frac{1}{8} = \frac{7}{8}$

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10 Probability Traps

1. "At least one" — use complement

Compute P(none) and subtract from 1. Much faster than listing all favorable cases.

2. AND means multiply; OR means add (if exclusive)

Common error: adding for AND events or multiplying for OR events.

3. Without replacement: denominator decreases

Each draw from a bag without replacement changes the sample space.

4. Independent vs dependent

Coin flips are independent. Drawing without replacement is dependent.

5. "Or" with overlap: subtract intersection

$P(A \cup B) = P(A)+P(B)-P(A \cap B)$. Forgetting to subtract double-counts.

6. P cannot exceed 1

If your calculation gives P > 1, you've made an error. Recheck.

7. Equally likely outcomes assumption

Only use favorable/total when all outcomes are equally likely (e.g., a fair die, not a weighted one).

8. Order vs combination in probability

Picking 2 from 5 in order vs not in order changes the sample space count.

9. Mutually exclusive ≠ independent

If A and B are mutually exclusive, knowing A happened tells you B didn't — they are NOT independent.

10. Probability of "exactly k"

Use the complement or direct counting, not just multiplying two probabilities.

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10 GMAT Practice Questions

Q1 PS Difficulty: 500

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

(B) 1/3. Numbers > 4: {5, 6}. Favorable = 2. Total = 6. $P = \frac{2}{6} = \frac{1}{3}$.

Q2 PS Difficulty: 500

A bag contains 5 red and 3 blue balls. What is the probability of drawing a red ball?

(B) 5/8. Red = 5, Total = 8. $P = \frac{5}{8}$.

Q3 PS Difficulty: 600

A coin is flipped 3 times. What is the probability of getting at least one head?

(D) 7/8. P(all tails) = $(1/2)^3 = 1/8$. P(at least one head) = $1 - 1/8 = 7/8$.

Q4 PS Difficulty: 600

Two dice are rolled. What is the probability that both show a 6?

(C) 1/36. $P(6) \times P(6) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

Q5 PS Difficulty: 650

From a standard 52-card deck, what is the probability of drawing a heart OR an ace?

(B) 16/52. $P(\text{heart}) = 13/52$. $P(\text{ace}) = 4/52$. $P(\text{ace of hearts}) = 1/52$. $P(\text{heart or ace}) = 13/52 + 4/52 - 1/52 = 16/52$.

Q6 PS Difficulty: 700

A bag has 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability both are red?

(A) 2/15. $P = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$.

Q7 DS Difficulty: 700

Is the probability that a randomly selected person from a group likes tea greater than 0.5?

(1) 60% of the group likes coffee.
(2) 40% of the group likes both tea and coffee.

(E) Neither sufficient. Neither statement alone nor together gives us the tea percentage directly. Someone might like neither, and the tea percentage is unknown without more info. (E).

Q8 PS Difficulty: 650

In a group of 50 people, 30 play tennis, 20 play golf, and 10 play both. If one person is chosen at random, what is the probability they play tennis or golf?

(B) 3/5. Tennis or golf = 30+20−10 = 40 people. P = 40/50 = 4/5. Wait: 40/50 = 4/5 = 0.8. That's choice (C). (C) 4/5 is correct.

Q9 PS Difficulty: 650

The probability that it rains on Monday is 0.4 and on Tuesday is 0.3. The days are independent. What is the probability it rains on at least one of the two days?

(C) 0.58. P(neither) = 0.6 × 0.7 = 0.42. P(at least one) = 1 − 0.42 = 0.58.

Q10 PS Difficulty: 700

A fair coin is flipped 4 times. What is the probability of getting exactly 2 heads?

(B) 3/8. Total outcomes = $2^4 = 16$. Ways to get exactly 2 heads: $\binom{4}{2} = 6$. $P = 6/16 = 3/8$.

Lesson Summary — Key Takeaways

P = favorable / total outcomes

Only valid when all outcomes are equally likely (fair coin, unweighted die, etc.).

"At least one" = 1 − P(none)

The complement strategy is the fastest path for at least-one problems.

AND = multiply; OR = add minus overlap

$P(A\cap B) = P(A)P(B)$ for independent. $P(A\cup B) = P(A)+P(B)-P(A\cap B)$.

Without replacement: update denominator

Each draw reduces total. After taking 1 of n items, next draw is from n−1.